Table of contents

For atoms whose valence states obey k=n+l ordering, the author assumes a Hamiltonian H=H^(k)+H^(l), where the first term has invariance group SO(4) and is perturbed by the second. The author derives matrix representations of operators which, with angular momentum, are constants of the motion and generate such an SO(4) group.

Symmetries and higher-order or generalised symmetries for the SKdV equation are constructed. Moreover by the introduction of graded potentials a non-local generalised symmetry is obtained, leading to the recursion operator for symmetries in a straightforward way, without making use of the bi-Hamiltonian structure.

The authors have analysed the equation of dispersive water waves within the formalism of a singular point analysis of the Painleve conjecture. It is interesting to observe that the two different similarity reductions lead to apparently different-looking ordinary differential equations with a similar singularity structure. While one is a pure Painleve equation, the other is not. On the other hand, a direct analysis of the PDE itself shows resonances at r=-1, 2, 3, 4 which contain those of the ODE. If, however, they adopt the exact procedure of Weiss et al. (1983) and try to truncate the expansion then it is seen that the set of equations obtained are not overdetermined. Thus it is not possible to derive the Lax pair via the Painleve analysis but a Backlund transformation can be written. The number of arbitrary coefficients is seen to be in conformity with the Cauchy-Kowalevskaya theorem.

It is shown that the singularity of the background scalar field, even if this field is topological trivial and/or localised in a finite region, will give rise to a fractional fermionic charge of the vacuum sector of a quantised Dirac field in (1+1) dimensions. The physical effect of the singularity of the background field is discussed.

An exact functional relation is established for the partition function of the Ising model with magnetic field formulated on a Kagome lattice of arbitrary size N. An exact solution found in a previous work, see ibid., vol.21, p.31, (1988), is a particular case of the result presented in this letter.

Numerical evidence is given to show that different aperiodic Ising models can have different universality classes.

The author considers a dissipative reactive system driven far from thermodynamic equilibrium and sustaining a locally attractive centre manifold beyond a symmetry-breaking instability. The author derives a functional equation for the renormalised diffusion pressure. This coefficient gives autocorrelations for intrinsic fluctuations and depends on the bifurcation parameter. The system is subject to the following realistic restrictions. (i) The temporal organisation lying beyond the instability can be accounted for by the existence of a centre manifold. (ii) The effective equilibrium energy averaged with respect to the stationary distribution about the centre manifold is independent of the bifurcation parameter.

The central charge for the integrable higher-spin XXZ model is derived, exploiting the exact Bethe ansatz solution by Babujian and Tsvelick (1986). For integer values of v, where gamma = pi /v is the anisotropy parameter (0< gamma < pi /2S), we find c=3S/(S+1), with S the magnitude of the spin. Hence, contrary to expectation, the U(1)-invariant critical point of the theory does not renormalise onto a free massless scalar field when S>1/2.

For an asymmetric version of the McCulloch-Pitts neural network (1943) and for Kauffman's infinite-range Boolean network model (1984), the time evolution of the Hamming distances between two different initial configurations are compared in the thermodynamic limit. It is shown that in both models phase transitions occur for corresponding values of the transition parameters and that their Hamming distances can have the same time evolution leading to quantitatively the same dynamics, as known from time-dependent Landau theory for phase transitions.

The Lie algebra associated with infinitesimal symmetry transformations of the equation of motion for a radiating charged particle embedded in the radiation field, i.e. the so-called Langevin equation, is determined. It is shown that the invariance group is a five-parameter group with a commutative three-parameter subgroup.

The author consider several non-linear equations which are known to be completely integrable by the method of inverse scattering transform, ranging from sine-Gordon and the Dodd-Bullough equation (1976) to the axisymmetric, stationary, Einstein-Maxwell equations. Such equations are known to be invariant under a fundamental Lie group of symmetry (G), such as SL(2) or SL(3) in the simpler cases. The author's treatment starts with the reformulation of the given equations as a system involving the scalar invariants of the Lie group (G) only; this scalar formulation is not only (G) invariant, but also manifestly invariant under the continuous group associated with the emergence of a spectral parameter, and is thus expected to be more fundamental and simpler. In all the cases that the author has considered, including the Einstein-Maxwell system, vector pseudopotentials of dimension not higher than three have been derived; Backlund transformations have been found, which generate multisolitons starting from the vacuum, and have the general (reciprocal) form: X'=1/X where X is a component of one of the vector (or tensor) pseudopotentials. The author also mentions the result that Kinnersley's formulation (1977) of the Ernst (vacuum) equation (1968) is formally equivalent to the fluid dynamical problem of unidimensional ideal gas flow; the Dodd-Bullough equation is equivalent to another particular case of gas motion.

The slow-amplitude modulation of a weakly nonlinear purely dispersive quasimonochromatic wave solution to a certain system of quasilinear partial differential equations is investigated. It is shown that, near the marginal state of modulational instability, and provided the system satisfies certain conditions, the complex amplitude of the wave is governed by a modified form of the nonlinear Schrodinger equation that involves higher-order nonlinearities.

Scattering by a one-dimensional potential V=V(x) which is constant outside (-a, a) (a>0) is investigated. Exact expressions for the transmission and reflection coefficients are obtained in terms of f₀(a), f₁(a), f'_O(a) and f'₁(a), where f₀ and f₁ are the solutions in (-a, a) satisfying the boundary conditions f₀(-a)=1, f₀'(-a)=0, f₁(-a)=0 and f₁'(-a)=1. These expressions are used to show that the first-order WKB approximation conserves particles. Numerical results are obtained from these expressions for the transmission and reflection probabilities by (i) using the WKB formulae and (ii) solving the time-independent Schrodinger equation numerically in (-a, a). The time T for the centre of a wavepacket to traverse the interval (-a, a) is also obtained in terms of f₀(a), f₁(a), f₀'(a) and f₁'(a), and some of its properties studied. In particular, it is shown that T takes the classical form in the classical limit.

The Runge-Lenz vector for a relativistic Coulomb problem in the quasipotential approach is found. The energy spectrum is obtained by an algebraic method. A hidden symmetry group of this problem is shown to be a group O(4) for mod E mod <1, SO(3, 1) for mod E mod >1 and a motion group of the three-dimensional Euclidean space for mod E mod =1.

The Hilbert space of one-photon states is introduced and used for an investigation of some basic principles of squeezing. Regarding the one-photon space as the underlying structure of the fields, the authors introduce parametrisations that simplify calculations considerably, and discuss how phase sensitivity, characteristic for squeezing, reveals itself in the one-photon space.

The problem of Aharonov-Bohm scattering on parallel flux lines of the same magnitude is solved exactly and the differential cross section is calculated.

Using the periodic solutions of the nonlinear Poisson-Boltzmann equation it is shown that in a many-particle Coulomb system a periodic localisation of charges may occur. The value of the charge density at which localisation commences is found. This density and the spacing of the localised charges are shown to be characteristic properties of the system under investigation. Their values characterise the self-organisation of the system. The results depend on the nonlinearity of the Poisson-Boltzmann equation and are not found if the linearised form is used.

The formal subdynamics approach to irreversibility in nonequilibrium statistical mechanics is cursorily reviewed, emphasis being placed on a recently developed time dependent formalism which generalises naturally from isolated systems to systems open to external influence. By means of a diagrammatic approach, the authors apply this theoretical framework to begin the study of gases and plasmas evolving in the presence of time dependent external fields. The general theorem in the dynamics of correlations is extended to include these cases and a corollary is deduced which enables one to eliminate from consideration a large class of diagrams in the thermodynamic limit. Finally, they discuss briefly some consequences of the formalism which may serve as avenues for the investigation of problems of physical interest.

Two algorithms for sequence extrapolation, due to van den Broeck and Schwartz (1979), and Bulirsch and Stoer (1964), are reviewed and critically compared. Applications to three-state and six-state quantum chains and to the (2+1)D Ising model show that the algorithm of Bulirsch and Stoer is superior, in particular if only very few finite-lattice data are available.

The generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously. These functions were found by Guttmann and Enting (1988). Their calculation is based on the series expansions up to the 64th order and is nonrigorous. The asymptotic form of the mean-squared radius of gyration of n-step convex polygons on the square lattice is determined and the critical exponent v is 1.

Learning in a generalised perceptron neutral network model is investigated by numerical simulation. It is found that the distribution of learning times is very broad and spreads out as the system size is increased. The mean number of steps, k, to learn a first-order task is found to increase with system size according to a power law (k) approximately N^alpha , alpha =1.86+or-0.05.

The authors exactly evaluate the character functions of SU(3) from the heat kernel. Their results reproduce the first Weyl formula. They evaluate the Weyl chamber for SU(3) and show its connection to the character functions.

From the eigenvalue equation H_lambda mod psi _n( lambda )>=E_n( lambda ) mod psi _n ( lambda )) where H_lambda identical to H₀+ lambda V one can derive an autonomous system of first-order differential equations for the eigenvalues E_n( lambda ) and the matrix elements V_mn( lambda ):=( psi _m( lambda ) mod V mod psi _n( lambda )) where lambda is the independent variable. To solve the dynamical system the authors need the initial values E_n( lambda =0) and mod psi _n( lambda =0)). Thus one finds the 'motion' of the energy levels E_n( lambda ). They derive the equations of motion for the extended case H_lambda =H₀+ lambda V₁+ lambda ²V₂ and give an application to a supersymmetric Hamiltonian.

The transfer matrix spectra of the magnetic hard squares along the multicritical line are analysed using the data for the strip width N up to 12. The operator content of the model is exactly the same as that of the Ashkin-Teller model. When N is odd, the spectrum is accounted for by the continuum theory in which the antiperiodic boundary condition is imposed to one of the Ising variables of the Ashkin-Teller model.